and the Laplacian on polygons

and the Laplacian on polygons

Luca Guido Molinari¹

Physics Department “Aldo Pontremoli”

Universit`a degli Studi di Milano and I.N.F.N. sez. Milano Via Celoria 16, 20133 Milano, Italy

Abstract: Several sums of Neumann series with Bessel and trigono- metric functions are evaluated, as finite sums of trigonometric func- tions. They arise from a generalization of the Neumann expansion of the eigenstates of the Laplacian in regular polygons. A simple accurate approximation of J₀(x) is found, on the interval [0, 2].

MSCS [2020]: Primary 33C10, Secondary 35J05.

Keywords: Bessel function, Neumann series, Laplace equation in poly- gon.

Resubmitted 5 apr 2021

Introduction

The ground state of the Laplace equation in a regular polygon with Dirichlet boundary conditions at the n sides, has a natural expression as a Neumann series of Bessel and trigonometric functions,

ψ_n(r, θ) = J₀(λ_nr) + 2X^∞

k=1h_k,nJ_kn(λ_nr) cos(knθ),

with coefficients h_k,n to be found and eigenvalue −λ²_n that scales with the area. For the equilateral triangle and the square, the solutions are known as sums of few trigonometric functions of the coordinates x = r cos θ and y = r sin θ. Such solutions have a corresponding Neumann expression [7]. For the square of area π:

J₀(r√

2π) + 2X^∞

k=1J_4k(r√

2π) cos(4kθ)

= ¹₂cos(x√

2π) + ¹₂cos(y√

2π) (1)

1Luca.Molinari@unimi.it

1

J₀(λ₃r) + 2X^∞

k=1

cos(kπ/2 − π/6)

cos(π/6) J_3k(λ₃r) cos(3kθ)

= ₃^√2₃sin(_3R^4π

3x + ^2π₃ ) −₃^√2₃[sin[_3R^2π

3(x + y√

3) −^2π₃ ]

−₃^√2₃sin[_3R^2π

3(x − y√

3) −^2π₃ ] (2) where, for area π, λ²₃ = 4π/√

3 and R₃ = ²₃p π√

3. In [7] I obtained a sum that generalizes the integrable cases n = 3, 4:

f_n(x, y) =J₀(r) + 2X^∞

k=1

cos[nk^3π₂ −_2n^π ]

cos(_2n^π) J_nk(r) cos(nkθ)

=1 n

Xⁿ⁻¹

`=0

cos[r cos(θ + ^2π_n`) + _2n^π] cos(_2n^π )

=1 n

Xⁿ⁻¹

`=0

cos[x cos(^2π_n`) − y sin(<sup>2π</sup><sub>n</sub>`) +_2n^π ]

cos(_2n^π ) (3)

For n → ∞ the Riemann sum in the second line is R2π 0

dt

2πcos(r cos t) = J0(r). For n = 2 it is f2(x, y) = cos x. For n = 6:

f6(x, y) =J0(r) + 2X^∞

k=1(−1)^kJ6k(r) cos(6kθ)

=¹₃cos x + ²₃cos(¹₂x) cos(

√ 3

2 y) (4)

The functions f_nare eigenfunctions of the Laplace operator with eigen- value −1 but, for n > 4, they no longer vanish on the boundary of a n-polygon. I only remark that the level curves fn(x, y) = C are closed around the origin (where fn= 1) up to a separatrix with n self- intersections, with values C₅ = −0.334909, C₆ = −1/3, C₇ = 0.19633, etc. The level lines and the separatrices for n = 6, 7 are shown in Fig.1.

The study of the Laplacian in polygons has a long history. The ground states beyond the square, n > 4, cannot be finite sums of trigonometric functions. They have been investigated analytically and numerically in 1/n expansion (see for example [7, 4, 3]).

In this paper I generalize the identity (3), and obtain a number of new formulas for Neumann series whose sums contain a finite num- ber of terms. For certain values of the parameters, they are identities that are found in the tables by Gradshteyn and Ryzhik [2], Prudnikov, Brychkov and Marichev [9], a recent paper by Al-Jarrah, Dempsey and Glasser [1], and two old papers by Takizawa and Kobayasi [10, 5]. In the last ones, the Neumann series appear as correlation functions for the heat flow in coupled harmonic oscillators.

A side result is an accurate approximation of J₀(x) on the interval [0, 2],

-5 0 5 -5

0 5

-5 0 5

-5 0 5

Figure 1. Left: the contour plot of the func- tion f₆ in eq.(4). Right: the separatrix f₆(x, y) =

−1/3 is a Kagom´e lattice. It can be written as 0 = cos(x/2)[cos(x/2) + cos(√

3y/2)]. Vertices are the dou- ble zeros.

-5 0 5

-5 0 5

-5 0 5

-5 0 5

Figure 2. Left: the contour plot of f₇ in eq.(3). Right:

the separatrix f₇(x, y) = −1.9633...

by three cosines (eq.18).

All figures and several checks were made with Wolfram Mathematica7.

The summation formula The source equation of various sums in this paper is:

+∞

X

k=−∞

J_kn+p(z)e^ikny = 1 n

n−1

X

`=0

e^{iz sin(y+2πn`)−ip(y+2πn`)} (5)

even n eq.(5) is eq.1 in [10]. Sums of this sort are tabulated only for n = 1, 2 in [9].

Proof. The result follows from the Fourier integral of a Bessel function of integer order. For z ∈ C, the sumP∞

k=−∞e^iknyJ_kn+p(z) is uniformly convergent in y by the bound |J±m(z)| ≤ C|z/2|^m/m! (Nielsen, see

§3.13 in [11]).

∞

X

k=−∞

e^iknyJ_kn+p(z) =

∞

X

k=−∞

e^ikny Z 2π

0

dθ

2πeiz sin θ−i(kn+p)θ

=

∞

X

k=−∞

e^ikny

n−1

X

j=0

Z ^2π_n(j+1)

2π nj

dθ

2πeiz sin θ−ipθe^−iknθ

=

n−1

X

j=0

" _∞ X

k=−∞

e^ikny Z ^2π_n

0

dθ

2πe^{iz sin(θ+2πnj)−ip(θ+2πnj)}e^−iknθ

#

The functions pn/2π e^ikny, k ∈ Z, are a complete orthonormal ba- sis in the Hilbert space L²(0, 2π/n). The infinite sum is the Fourier representation of (1/n) exp[iz sin(y + ^2π_nj) − ip(y + ^2π_nj)]. 

In the following I give some examples.

1. The case p = 0 and y = ^π₂ + α is an extension with angle α of the equations 19 and 20 in [1], where α = 0. With J−m(z) = (−)^mJ_m(z):

J₀(z) + 2

∞

X

k=1

e^iknπ2J_kn(z) cos(knα) = 1 n

n−1

X

`=0

e^{iz cos(α+2πn`)} (6)

For n = 1, separation of even and odd parity terms in z gives the Jacobi expansions (eqs. 5.7.10.4 and 5 in [9]):

J₀(z) + 2

∞

X

k=1

(−)^kJ_2k(z) cos(2kα) = cos(z cos α) (7)

∞

X

k=0

(−)^kJ_2k+1(z) cos[(2k + 1)α] = ¹₂sin(z cos α) (8)

If α is replaced by α + π/2 they are eqs. 8.514.5 and 6 in [2] and 10.4, 10.5 in [6]:

J₀(z) + 2

∞

X

k=1

J_2k(z) cos(2kα) = cos(z sin α) (9)

∞

X

k=0

J_2k+1(z) sin[(2k + 1)α] = ¹₂sin(z sin α) (10)

1.1. For n replaced by 2n, eq.(6) is:

J0(z) + 2

∞

X

k=1

(−)^knJ2kn(z) cos(2knα) = 1 n

n−1

X

`=0

cos[z cos(α + ^π_n`)] (11) where the finite sum was reduced by noting that terms ` and n + ` are the same.

- The value y = ^π₂ yields eq.(23) in [1].

- For n = 1 the derivative of (11) in α = ^π₄ is:

2J₂(z) − 6J₆(z) + 10J₁₀(z) − 14J₁₄(z) + ... = z

√2 4 sin(z

√2

2 ) (12) - For n = 2 eq.(11) becomes

J₀(z) + 2X^∞

k=1J_4k(z) cos(4kα) = 1

2[cos(z sin α) + cos(z cos α)]. (13) The values α = 0,^π₄ give eqs.5.7.1.19 in [9]. The derivative is:

∞

X

k=1

kJ_4k(z) sin(4kα) = z

16[sin(z sin α) cos α − sin(z cos α) sin α], (14) further, the expansion in small α gives:

∞

X

k=1

k²J_4k(z) = z

64(z − sin z) (15)

- Case n = 3 is eq.(4), α = 0 gives eq. 5.7.1.21 in [9].

1.2. If n is replaced by 2n + 1, eq.(6) is:

J₀(z) + 2

∞

X

k=1

e^i(2n+1)kπ2J_(2n+1)k(z) cos[(2n + 1)kα] = 1 2n + 1

2n

X

`=0

e^{iz cos(α+}

2π 2n+1`)

J₀(z) + 2

∞

X

k=1

(−)^kJ_(4n+2)k(z) cos[(4n + 2)kα]

= 1

2n + 1

2n

X

`=0

cos[z cos(α +_2n+1^2π `)] (16)

2

∞

X

k=0

(−)^n+kJ(2n+1)(2k+1)(z) cos[(2n + 1)(2k + 1)α]

= 1

2n + 1

2n

X

`=0

sin[z cos(α +_2n+1^2π `)] (17) The first equation with n = 1 and α = π/12, offers a simple approx- imation of J0(x) in 0 ≤ x ≤ 2 with an error || ≤ 4 × 10⁻⁹

J₀(x) = ¹₃[cos(x^√1

2) + cos(x

√ 3−1 2√

2 ) + cos(x

√ 3+1 2√

2 )] +  (18) At j_0,1 ≈ 2.404 (first zero of J₀(x)) the error is 3.4 × 10⁻⁸.

Examples of the second equation, (17), are:

∞

X

k=0

(−)^kJ_6k+3(z) cos[(6k + 3)α] = −1 6

2

X

`=0

sin[z cos(α +^2π₃ `)] (19)

∞

X

k=0

(−)^kJ_10k+5(z) cos[(10k + 5)α] = 1 10

4

X

`=0

sin[z cos(α +^2π₅ `)] (20) The first one with α = π is eq.22 in [1].

Both sums are eigenfunctions of the Laplacian with eigenvalue λ = −1 (see Figs. 3,4). The finite sum in (20), with z = r, x = r cos α and y = r sin α, is:

f (x, y) = ₁₀¹ sin x − ¹₅sin(x cos^π₅) cos(y sin^π₅)

+¹₅sin(x cos^2π₅ ) cos(y sin^2π₅ ) (21) 1.3. Eq. (5) with p = 0 is multiplied by exp(iβ), β real, and the real part is taken. The left hand side becomes:

J₀(x) cos β +X^∞

k=1J_kn(x) Re[e^iβ(e^ikny + e^−ikn(y+π))]

= J₀(x) cos β + 2X^∞

k=1cos(β − kn^π₂)J_kn(x) cos[kn(y + ^π₂)]

The identity (3) is obtained, when β = _2n^π and y + ^π₂ = θ + π.

-5 0 5 -5

0 5

-5 0 5

-5 0 5

Figure 3. Contour plot of the sum (19). The function is the ground state of the Laplacian, vanishing on the boundary of equilateral triangles (no nodal lines) and is an excited state of the hexagon (the right figure is the contour for the value zero).

-5 0 5

-5 0 5

-5 0 5

-5 0 5

Figure 4. Left: contour plot of the sum (20), i.e the function f in (21). Right: the contour f = 0 is very close to the circumference of radius j_5,1 (the first zero of J₅), where |f (x, y)| ≤ 0.001.

2. Parseval’s identity is applied to (5):

X

k∈Z

J_kn+p² (x) = 1 n²

X

k`

e^{ip2πn(k−`)} Z 2π

0

dy

2πe^{ix sin(y−πn}(k−`))−ix sin(y+π n(k−`))

= 1 n²

X

k`

e^{ip2πn(k−`)} Z 2π

0

dy

2πe−i2x cos y sin(π n(k−`))

= 1 n²

X

k`

e<sup>ip2πn(k−`)</sup>J<sub>0</sub>[2x sin(<sup>π</sup><sub>n</sub>(k − `))]

= 1 n + 2

n²

n−1

X

k=1

k cos(^2π_nkp)J₀(2x sin^πk_n)

X

k∈Z

J_kn+p² (x) = 1 n + 1

n

n−1

X

k=1

cos(^2π_nkp)J₀(2x sin^πk_n) (22) The left-hand side is Jp(x)²+P∞

k=1[J_kn+p² (x) + J_kn−p² (z)].

2.1. If n → 2n and p = 0, the sum (22) is amenable to eq.29 in [1]:

J₀²(x) + 2

∞

X

k=1

J_2kn² (x) = 1 2n + 1

2nJ0(2x) + 1 n

n−1

X

k=1

J0(2x cos_2n^πk) (23) - If n → 2n and p = n in (22), with simple steps one obtains:

∞

X

k=0

J_(2k+1)n² (x) = 1

4n + (−)ⁿ

4n J₀(2x) + 1 2n

n−1

X

`=1

(−1)^{`</sup>J<sub>0</sub>(2x sin<sub>2n</sub><sup>π`}) (24) Examples:

X^∞

k=0J_2+4k² (x) = ¹₈ +₈¹J₀(2x) − ¹₄J₀(x√

2) (25)

X^∞

k=0J_3+6k² (x) = ₁₂¹ − ¹₆J₀(x) −₁₂¹J₀(2x) + ¹₆J₀(x√

3) (26) 3. In eq.(5) the variable y is shifted to y+2t. The equation is multiplied by e^{iz0sin y−iqy} and integrated in y:

+∞

X

k=−∞

J_p+kn(z)J_q−kn(z⁰)e^i(kn+p)2t

=1 n

n−1

X

`=0

e^−ip2πn` Z 2π

0

dy

2πeiz sin(y+2t+2π

n`)+iz⁰sin y−i(p+q)y

(27) In the integral, the shift y to y − t − ^π_n` changes the exponent to i(z + z<sup>0</sup>) sin y cos(t +<sup>π</sup><sub>n</sub>`) + i(z − z⁰) cos y sin(t +^π_n`) − i(p + q)(y − t −<sup>π</sup><sub>n</sub>`) 3.1. With z = z⁰ we obtain eq.1 in [5]:

+∞

X

k=−∞

Jp+kn(z)Jq−kn(z)e^2iknt = 1 n

n−1

X

`=0

e<sup>−i(p−q)(t+πn`)</sup>Jp+q[2z cos(t +<sup>π</sup><sub>n</sub>`)]

(28) For n = 1, with a shift of the index k and renaming of parameter, it is eq.8.530 in [2].

For n = 2, p = q and t = π/4 it is eq. 5.7.11.25 [9].

3.2. Eq.(27) with p = q = 0 and t = 0 is:

J₀(z)J₀(z⁰) + 2

∞

X

k=1

(−)^knJ_kn(z)J_kn(z⁰) = 1 n

n−1

X

`=0

Z 2π 0

dy

2πe^{iz sin(y+2πn`)+iz0sin y} (29) For n = 1, 2 they are eqs. 5.7.11.1 and 5.7.11.18 in [9]; if also z = z⁰ they are eqs. 31, 32 in [1].

A new example is:

∞

X

k=1

J_4k(x)J_4k(y) = ¹₈[J₀(x + y) + J₀(x − y) − 4J₀(x)J₀(y)]

+¹₄J0(p

x²+ y²) (30) 4. Multiplication of (5) by exp(−ay) (a > 0) with p = 0 and n = 1, and integration on R⁺ give:

∞

X

k=−∞

J_2k(z) a

a²+ 4k² + 2i

∞

X

k=0

J_2k+1(z) 2k + 1 a²+ (2k + 1)² =

Z ∞ 0

dy eiz sin y−ay

The last integral is done by expanding exp(iz sin y) and using integrals eqs.3.895.1 and 3.895.4 [2]. The even and odd terms are:

1

a²J0(z) + 2

∞

X

k=1

J_2k(z) a²+ 4k² =

∞

X

k=0

(−)^kz^2k

a²(a²+ 4)...(a²+ 4k²) (31)

∞

X

k=0

J2k+1(z) 2(2k + 1) a²+ (2k + 1)² =

∞

X

k=0

(−)^kz^2k+1

(a²+ 1)(a²+ 9)...(a²+ (2k + 1)²) (32) For a = iν the equations are expansions of the Lommel functions s−1,ν(z) and s_0,ν(z) in series of Bessel functions (eq. 11.9.7 in [8]).

More and more identities can be obtained by derivation, or integra- tion with functions. Here I limited myself to some examples.

Data availability. Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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